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Properties

Label 2-93e2-1.1-c1-0-73
Degree 22
Conductor 86498649
Sign 11
Analytic cond. 69.062669.0626
Root an. cond. 8.310398.31039
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.23·2-s − 0.481·4-s + 1.54·5-s − 3.80·7-s + 3.05·8-s − 1.90·10-s + 3.76·11-s − 2.63·13-s + 4.68·14-s − 2.80·16-s − 3.77·17-s + 6.09·19-s − 0.744·20-s − 4.63·22-s − 0.909·23-s − 2.61·25-s + 3.24·26-s + 1.83·28-s + 6.80·29-s − 2.66·32-s + 4.65·34-s − 5.87·35-s − 1.81·37-s − 7.51·38-s + 4.72·40-s − 0.337·41-s + 3.88·43-s + ⋯
L(s)  = 1  − 0.871·2-s − 0.240·4-s + 0.691·5-s − 1.43·7-s + 1.08·8-s − 0.602·10-s + 1.13·11-s − 0.730·13-s + 1.25·14-s − 0.701·16-s − 0.915·17-s + 1.39·19-s − 0.166·20-s − 0.988·22-s − 0.189·23-s − 0.522·25-s + 0.636·26-s + 0.346·28-s + 1.26·29-s − 0.470·32-s + 0.797·34-s − 0.993·35-s − 0.298·37-s − 1.21·38-s + 0.747·40-s − 0.0526·41-s + 0.592·43-s + ⋯

Functional equation

Λ(s)=(8649s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8649s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 86498649    =    32⋅3123^{2} \cdot 31^{2}
Sign: 11
Analytic conductor: 69.062669.0626
Root analytic conductor: 8.310398.31039
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8649, ( :1/2), 1)(2,\ 8649,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.88651401860.8865140186
L(12)L(\frac12) ≈\approx 0.88651401860.8865140186
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
31 1 1
good2 1+1.23T+2T2 1 + 1.23T + 2T^{2}
5 1−1.54T+5T2 1 - 1.54T + 5T^{2}
7 1+3.80T+7T2 1 + 3.80T + 7T^{2}
11 1−3.76T+11T2 1 - 3.76T + 11T^{2}
13 1+2.63T+13T2 1 + 2.63T + 13T^{2}
17 1+3.77T+17T2 1 + 3.77T + 17T^{2}
19 1−6.09T+19T2 1 - 6.09T + 19T^{2}
23 1+0.909T+23T2 1 + 0.909T + 23T^{2}
29 1−6.80T+29T2 1 - 6.80T + 29T^{2}
37 1+1.81T+37T2 1 + 1.81T + 37T^{2}
41 1+0.337T+41T2 1 + 0.337T + 41T^{2}
43 1−3.88T+43T2 1 - 3.88T + 43T^{2}
47 1−1.18T+47T2 1 - 1.18T + 47T^{2}
53 1−2.34T+53T2 1 - 2.34T + 53T^{2}
59 1+7.77T+59T2 1 + 7.77T + 59T^{2}
61 1+2.72T+61T2 1 + 2.72T + 61T^{2}
67 1+7.42T+67T2 1 + 7.42T + 67T^{2}
71 1−5.09T+71T2 1 - 5.09T + 71T^{2}
73 1−5.39T+73T2 1 - 5.39T + 73T^{2}
79 1+9.73T+79T2 1 + 9.73T + 79T^{2}
83 1−8.39T+83T2 1 - 8.39T + 83T^{2}
89 1+5.09T+89T2 1 + 5.09T + 89T^{2}
97 1−10.9T+97T2 1 - 10.9T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.76102595848559008542190342946, −7.11062479679026086816998470195, −6.51320188548594663624822262760, −5.91422868941683068232596470587, −4.98996217672024392502305748755, −4.22866319465634262408542711787, −3.41206003910733068899875080636, −2.52448557110213957051061563805, −1.52878373452986343005696448779, −0.54685361318613941626128335031, 0.54685361318613941626128335031, 1.52878373452986343005696448779, 2.52448557110213957051061563805, 3.41206003910733068899875080636, 4.22866319465634262408542711787, 4.98996217672024392502305748755, 5.91422868941683068232596470587, 6.51320188548594663624822262760, 7.11062479679026086816998470195, 7.76102595848559008542190342946

Graph of the ZZ-function along the critical line